Fourier Transform Calculator

Explore common continuous Fourier transform pairs with convention toggles, magnitude and phase plots, and signal previews.

Results are calculated automatically as you enter data.

Operation Function family Convention Amplitude A Width or sigma Angular frequency Ω

Piecewise constant intervals

	Start	End	Value

Each row represents a constant value from start to end.

Transform pair notes

Forward transform Choose a signal.

Time domain

Frequency magnitude

Frequency phase

Disclaimer: All tools on this site are intended for educational and informational purposes only. The analytical formula follows the selected convention exactly; graphs are sampled visualizations. Verify the model and units before important use. See our Terms of Use.

Assumptions and limitations

This calculator displays exact analytical formulas for its preset transform pairs and numerically samples regular functions only for plotting and tables. Pure sinusoids are shown as Dirac impulses, not finite-width peaks. Inputs are validated for supported precision and numeric range; unreliable results are rejected. Use of this calculator is subject to our Terms of Use.

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What Is a Continuous Fourier Transform?

A continuous Fourier transform is a way to describe a signal or function by its frequency content instead of by its time-domain shape. The original function, usually written as $f(t)$ or $x(t)$, tells you how the signal changes over time. Its Fourier transform, usually written as $F(\omega)$ or $X(\omega)$, tells you how much of each angular frequency $\omega$ is present.

The main idea is that many signals can be understood as combinations of complex sinusoids. A smooth wave, a short pulse, a decaying signal, and a piecewise-constant shape can all be studied by asking: which frequencies are needed to build this signal, and with what magnitude and phase?

For the angular-frequency convention used here, the forward transform is:

$$ F(\omega)=\int_{-\infty}^{\infty} f(t)e^{-i\omega t}\,dt $$

A matching inverse transform is:

$$ f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(\omega)e^{i\omega t}\,d\omega $$

In this convention, $\omega$ is angular frequency in radians per unit time. If a hertz-style frequency axis $f$ is used instead, the relationship is:

$$ \omega=2\pi f $$

In hertz form, the matching pair is (F(f)=\int f(t)e^{-i2\pi ft}\,dt) and (f(t)=\int F(f)e^{i2\pi ft}\,df). The unitary angular convention instead places (1/\sqrt{2\pi}) in both directions. The calculator rewrites each selected analytical pair accordingly.

The output is usually complex, so it is often easier to interpret through magnitude and phase. If

$$ F(\omega)=R(\omega)+iI(\omega) $$

then:

$$ \lvert F(\omega)\rvert=\sqrt{R(\omega)^2+I(\omega)^2} $$

and

$$ \phi(\omega)=\operatorname{atan2}\left(I(\omega),R(\omega)\right) $$

Magnitude answers “how much of this frequency is present?” Phase answers “how is that frequency shifted relative to the time origin?”

Why Fourier Transforms Matter

Fourier transforms are useful because many problems become simpler in the frequency domain. In signal processing, a filter can be understood by how it changes different frequencies. In engineering and physics, differential equations often become algebraic relationships after a Fourier transform. In communications, audio, vibration analysis, optics, and control systems, the frequency spectrum helps explain what a time-domain graph alone may hide.

The same time-domain signal can have several important frequency-domain features:

A short pulse usually spreads across many frequencies.
A wide, smooth signal usually concentrates more tightly near low frequencies.
A time shift changes phase without changing the magnitude spectrum.
A causal signal, such as a one-sided exponential, commonly has nonzero phase.
A pure sine or cosine corresponds to ideal frequency impulses rather than ordinary finite-width bumps.

This is why a Fourier transform is not just a formula. It is a different way of seeing the same signal.

Key Terms to Know

Time domain: The original view of the signal as a function of $t$.
Frequency domain: The view of the signal as a function of frequency, usually $\omega$.
Angular frequency: Frequency measured in radians per unit time. It is written as $\omega$.
Hertz-style frequency: Frequency measured in cycles per unit time. It is often written as $f$ and is related to angular frequency by $\omega=2\pi f$.
Transform pair: A time-domain signal and its matching frequency-domain transform.
Magnitude spectrum: The absolute value $\lvert F(\omega)\rvert$ of the complex transform.
Phase spectrum: The angle of the complex transform, usually measured in radians.
Sinc function: In this calculator, $\operatorname{sinc}(x)=\frac{\sin x}{x}$ with $\operatorname{sinc}(0)=1$.
Dirac delta: An ideal impulse used to represent energy concentrated at an exact frequency or time location.
Unit step: A signal often written as $u(t)$ that is zero before $t=0$ and active after $t=0$.
Unitary convention: A symmetric Fourier convention that applies $\frac{1}{\sqrt{2\pi}}$ in both the forward and inverse angular-frequency transforms.
Piecewise-constant signal: A signal made from constant values over specified intervals.

How Continuous Fourier Transforms Work

The Fourier transform measures how strongly $f(t)$ matches each complex sinusoid $e^{i\omega t}$. The integral multiplies the signal by $e^{-i\omega t}$ and accumulates the result over time. When the signal contains a strong component at that frequency, the accumulation is large. When positive and negative oscillations cancel, the magnitude is small.

Different signal families produce recognizable transform patterns.

Rectangular pulse

A centered rectangular pulse with amplitude $A$ and width $W$ is:

$$ f(t)= \begin{cases} A, & \lvert t\rvert\le \frac{W}{2}\\ 0, & \text{otherwise} \end{cases} $$

Its transform is:

$$ F(\omega)=AW\operatorname{sinc}\left(\frac{\omega W}{2}\right) $$

where:

$$ \operatorname{sinc}(x)=\frac{\sin x}{x},\qquad \operatorname{sinc}(0)=1 $$

The value at $\omega=0$ is the area of the pulse, $AW$. A wider pulse has a narrower main lobe in frequency; a narrower pulse spreads across more frequencies.

Gaussian signal

A Gaussian with amplitude $A$ and width parameter $\sigma$ is:

$$ f(t)=Ae^{-\frac{t^2}{2\sigma^2}} $$

Its transform is also Gaussian:

$$ F(\omega)=A\sigma\sqrt{2\pi}\,e^{-\frac{\sigma^2\omega^2}{2}} $$

This is one of the most important Fourier transform pairs. A Gaussian remains Gaussian under the transform, but the width relationship reverses: a wider Gaussian in time produces a narrower Gaussian in frequency.

One-sided exponential

A causal, one-sided exponential with amplitude $A$ and decay rate $\alpha$ is:

$$ f(t)=Ae^{-\alpha t}u(t),\qquad \alpha>0 $$

Its transform is:

$$ F(\omega)=\frac{A}{\alpha+i\omega} $$

Equivalently, its real and imaginary parts are:

$$ R(\omega)=\frac{A\alpha}{\alpha^2+\omega^2} $$

$$ I(\omega)=-\frac{A\omega}{\alpha^2+\omega^2} $$

Because the signal starts at $t=0$ instead of being even around the origin, its transform generally has meaningful phase variation.

Sine and cosine

Pure sinusoids are ideal frequency components. With angular rate $\Omega$, the cosine transform pair is:

$$ A\cos(\Omega t)\quad\longleftrightarrow\quad \pi A\left[\delta(\omega-\Omega)+\delta(\omega+\Omega)\right] $$

The sine transform pair is:

$$ A\sin(\Omega t)\quad\longleftrightarrow\quad -i\pi A\left[\delta(\omega-\Omega)-\delta(\omega+\Omega)\right] $$

The important point is that an ideal sinusoid does not occupy a smooth band of frequencies. It is concentrated at exact positive and negative angular frequencies.

Piecewise-constant signal

For a piecewise-constant signal with value $c_k$ over each interval $[a_k,b_k]$, the transform can be built by adding the transform contribution of each interval:

$$ F(\omega)=\sum_k c_k\frac{e^{-i\omega a_k}-e^{-i\omega b_k}}{i\omega} $$

At $\omega=0$, the expression is evaluated by area instead of by dividing by zero:

$$ F(0)=\sum_k c_k(b_k-a_k) $$

This area rule is a useful way to understand the zero-frequency value. It represents the total signed area under the time-domain signal.

Examples of Fourier Transforms in Practice

Example 1: Rectangular pulse and sinc spectrum

Suppose a centered rectangular pulse has amplitude $A=2$ and width $W=3$:

$$ f(t)=2 \quad \text{for} \quad \lvert t\rvert\le \frac{3}{2} $$

The transform is:

$$ F(\omega)=AW\operatorname{sinc}\left(\frac{\omega W}{2}\right) $$

Substitute $A=2$ and $W=3$:

$$ F(\omega)=6\operatorname{sinc}\left(\frac{3\omega}{2}\right) $$

At zero frequency:

$$ F(0)=6 $$

That value is the area of the pulse:

$$ \text{area}=2\times 3=6 $$

The first zeros occur when:

$$ \frac{3\omega}{2}=\pm \pi $$

so:

$$ \omega=\pm \frac{2\pi}{3} $$

The sinc shape shows a strong low-frequency lobe with oscillating side lobes.

Example 2: One-sided exponential and phase

Suppose a causal exponential has amplitude $A=4$ and decay rate $\alpha=2$:

$$ f(t)=4e^{-2t}u(t) $$

The transform is:

$$ F(\omega)=\frac{4}{2+i\omega} $$

Using real and imaginary parts:

$$ R(\omega)=\frac{8}{4+\omega^2} $$

$$ I(\omega)=-\frac{4\omega}{4+\omega^2} $$

At $\omega=0$:

$$ F(0)=\frac{4}{2}=2 $$

At $\omega=2$:

$$ R(2)=1, \qquad I(2)=-1 $$

The magnitude is:

$$ \lvert F(2)\rvert=\sqrt{1^2+(-1)^2}=\sqrt{2}\approx 1.4142 $$

The phase is:

$$ \phi(2)=\operatorname{atan2}(-1,1)=-\frac{\pi}{4} $$

This example shows why phase matters. The magnitude tells you the strength of the component at $\omega=2$, while phase tells you how the complex value is angled.

Example 3: Piecewise interval and the zero-frequency edge case

Suppose a signal equals $3$ from $t=0$ to $t=2$ and is zero elsewhere. This is a single piecewise-constant interval with $a=0$, $b=2$, and $c=3$.

For $\omega\ne 0$:

$$ F(\omega)=3\frac{e^{-i\omega\cdot 0}-e^{-i\omega\cdot 2}}{i\omega} $$

Since $e^0=1$:

$$ F(\omega)=3\frac{1-e^{-i2\omega}}{i\omega} $$

At $\omega=0$, use the interval area:

$$ F(0)=3(2-0)=6 $$

This avoids a false division-by-zero problem. The transform is continuous in the limiting sense, and the zero-frequency value is the signed area of the interval.

How to Interpret the Result

The transform formula describes the ideal analytical transform pair for the selected signal model. It is the main mathematical result, but it should be read together with the magnitude spectrum, phase spectrum, graph range, and frequency convention.

The frequency variable is $\omega$ unless the hertz convention is selected. In hertz mode, the calculator rewrites the analytical result using the kernel $e^{-i2\pi ft}$, substitutes $\omega=2\pi f$, and labels values in cycles per unit time. In unitary mode, the displayed formula and numerical values both include the symmetric normalization.

The magnitude curve or magnitude column is $\lvert F(\omega)\rvert$. Larger magnitude means a stronger contribution at that frequency. For a rectangular pulse, the magnitude follows a sinc-shaped envelope. For a Gaussian, the magnitude is Gaussian. For a one-sided exponential, the magnitude decreases as $\lvert\omega\rvert$ grows. For pure sine and cosine signals, ideal frequency content appears as impulses.

The phase curve or phase column is measured in radians. Phase is near zero for many real, even signals, such as a centered rectangular pulse or a centered Gaussian. Phase becomes more important when the signal is odd, shifted, causal, or complex-valued. A one-sided exponential is causal, so its transform has a real part and an imaginary part, which produces phase variation.

For piecewise-constant inputs, the result is the sum of the interval contributions. If intervals overlap, their values add in the time-domain model. If an interval value is negative, it contributes negative area and can change both magnitude and phase.

For sine and cosine inputs, the frequency plots use vertical impulse stems at the exact positive and negative frequencies. Their table reports the complex distribution weights; the stems do not represent an ordinary finite bandwidth.

Common Mistakes and Misconceptions

Confusing angular frequency with hertz. Angular frequency $\omega$ is measured in radians per unit time, while hertz-style frequency $f$ is measured in cycles per unit time. The conversion is $\omega=2\pi f$.

Expecting every transform to be a regular curve. Pure sine and cosine signals transform into ideal impulses. The calculator therefore uses exact impulse stems and distribution weights instead of drawing artificial finite-width peaks.

Forgetting that width must be positive. Rectangular and Gaussian width parameters must be greater than zero. A zero or negative width does not define the intended signal model.

Forgetting that rate must be positive. The one-sided exponential uses a positive decay rate. A sine or cosine uses positive angular frequency in angular modes and positive cycles-per-time frequency in hertz mode. Switching conventions converts the displayed sinusoid frequency so the physical signal is preserved.

Using the wrong Gaussian width meaning. The Gaussian model here uses a sigma-like width parameter in $e^{-t^2/(2\sigma^2)}$. It is not the same as full width at half maximum unless you convert between the definitions.

Interpreting phase without considering sign. A negative amplitude can shift phase by $\pi$ compared with a positive-amplitude version. Phase can also jump where magnitude is near zero.

Assuming the inverse direction accepts an arbitrary spectrum. The direction control reverses the selected analytical pair and makes its time-domain member the result. It is not a custom numerical inverse-transform input.

Assuming graph ranges are manually configurable. Plot ranges automatically adapt to the selected family and its parameters or interval support. There are no manual range controls, so read the axis labels before comparing two models.

Entering invalid piecewise intervals. For a piecewise interval, the end must be greater than the start. Imported interval text should follow the expected interval-and-value pattern, such as $a..b:\text{ value}$, or contain exactly three finite numbers per row: start, end, and value.

When to Use Continuous Fourier Transforms

Use a continuous Fourier transform when you want to understand an idealized continuous-time signal in terms of its frequency content.

Common use cases include:

comparing how different pulse widths affect bandwidth;
studying why a Gaussian remains Gaussian in the frequency domain;
analyzing the magnitude and phase of a causal exponential response;
understanding why ideal sinusoids correspond to frequency impulses;
building a transform for a piecewise-constant signal by adding interval contributions;
checking how transform conventions change scaling or the frequency axis;
learning the connection between a time-domain waveform and its spectrum.

For sampled data, finite records, or numerical spectrum estimation, a discrete Fourier transform or fast Fourier transform is usually the more appropriate tool. A continuous Fourier transform describes an ideal continuous model.

Limitations and Things to Keep in Mind

Fourier transforms are exact for the mathematical signal model being used, but the model may be an idealization. Real measurements are sampled, finite, noisy, and often windowed. That means a real-world spectrum can differ from an ideal continuous transform pair.

The formulas here use a continuous-time transform with the $e^{-i\omega t}$ kernel before optional convention scaling. Other books, software packages, and courses may place $2\pi$ factors differently or use $j$ instead of $i$ for the imaginary unit. Always check the convention before comparing formulas.

The unitary angular convention uses $\frac{1}{\sqrt{2\pi}}$ in both transform directions. The hertz convention uses $e^{-i2\pi ft}$ forward and $e^{i2\pi ft}$ inverse. Formulas, tables, impulses, and axes all follow the selected convention.

The supported signal families are preset analytical models and piecewise-constant signals. Arbitrary custom functions are not part of this calculator workflow. For functions outside these families, use a symbolic transform, a numerical integration method, or a discrete transform method as appropriate.

The rectangular and Gaussian models require positive width. The exponential and sinusoid models require positive rate or frequency. Numeric inputs are limited to 15 significant digits. Piecewise signals support at most 200 finite, ordered intervals, and pasted input is limited to 50,000 characters.

Pure sine and cosine spectra contain ideal Dirac delta impulses. The displayed stems show impulse locations and weights only; they should not be interpreted as physical bandwidth unless a window, finite duration, or measurement process has also been modeled.

Tables may shorten values for display but never deliberately replace a finite nonzero value with zero. Very large or small values use scientific notation. Regular-function tables sample five points across the adaptive plotted range and are snapshots rather than complete descriptions.

How to Use This Calculator

Choose the transform view or operation.
Select a signal family: rectangular pulse, Gaussian, one-sided exponential, cosine, sine, or piecewise constant.
Choose the transform convention. Use angular frequency $\omega$, symmetric unitary angular scaling, or hertz frequency $f$. Sine and cosine frequency inputs are converted when switching between angular and hertz conventions so the same physical signal is retained.
Enter amplitude for non-piecewise signals.
Enter width for rectangular or Gaussian signals, or enter rate for one-sided exponential, cosine, or sine signals.
For a piecewise-constant signal, enter each interval start, interval end, and value. Add rows or import interval text when needed.
Review the transform formula, summary notes, time-domain graph, frequency-domain graph, and frequency table.
Use the magnitude and phase explanations above to interpret the result rather than reading the formula alone.

Frequently Asked Questions

Is a Fourier transform the same as an FFT?

No. A continuous Fourier transform is an analytical transform for a continuous-time function. An FFT is an efficient algorithm for computing a discrete Fourier transform from sampled data. They are related ideas, but they apply to different types of inputs.

Why are there negative frequencies?

Complex sinusoids use both positive and negative angular frequencies. For real-valued signals, the negative-frequency side is related to the positive-frequency side by symmetry. Keeping both sides makes the mathematics of magnitude, phase, modulation, and inverse transforms consistent.

Why does a rectangular pulse transform into a sinc shape?

A rectangular pulse is sharply cut off in time. Sharp edges require many sinusoidal components to represent them, which creates oscillating side lobes in frequency. The sinc function is the analytical result of integrating a constant pulse over a finite interval.

Why does a Gaussian stay Gaussian after the Fourier transform?

The Gaussian has a special exponential-square form that remains in the same family under the Fourier transform. The width changes inversely between the time and frequency domains: a wider Gaussian in time becomes narrower in frequency.

Why do sine and cosine transforms show impulses?

A pure sine or cosine has exactly one positive and one negative frequency. In the ideal continuous transform, that concentrated content is represented by Dirac delta impulses. The calculator shows these as exact stems with complex weights, not as ordinary finite-width curves.

What does the zero-frequency value mean?

For many ordinary integrable signals, $F(0)$ equals the total signed area under the time-domain function. For example, a rectangular pulse has $F(0)=AW$, and a piecewise-constant signal has $F(0)=\sum_k c_k(b_k-a_k)$. Positive and negative areas can cancel.

What does phase tell me?

Phase tells you the angle of the complex transform value at each frequency. It is especially important for shifted, causal, or odd signals. Two signals can have similar magnitude spectra but different phase spectra, which means they can reconstruct to different time-domain shapes.

Sources and References

Books

Ronald N. Bracewell. The Fourier Transform and Its Applications. 3rd ed., McGraw Hill, 2000. Relevant topics: Fourier transform definition, transform pairs, sinc behavior, impulse interpretation, and applications. ISBN 9780073039381. Google Books listing.
Alan V. Oppenheim, Alan S. Willsky, and S. Hamid Nawab. Signals & Systems. 2nd ed., Prentice Hall, 1997. Relevant topics: continuous-time Fourier transform, Fourier transform properties, magnitude and phase, and signal-transform pairs. ISBN 9780138147570. Google Books listing.

Online and Official Sources

MIT OpenCourseWare. “Lecture 8: Continuous-Time Fourier Transform.” Signals and Systems, RES.6-007, Spring 2011. Instructor: Alan V. Oppenheim. Accessed June 28, 2026. MIT OpenCourseWare.
Richard Baraniuk et al. “8: Continuous Time Fourier Transform (CTFT),” including “8.2: Continuous Time Fourier Transform,” “8.3: Common Fourier Transforms,” and “8.4: Properties of the CTFT.” Signals and Systems, Engineering LibreTexts. Accessed June 28, 2026. Engineering LibreTexts.
NIST Digital Library of Mathematical Functions. “§1.14 Integral Transforms.” Accessed June 28, 2026. NIST DLMF.